Question

1. here is the recursive definition of a sequence: ( f_1 = 60 ), ( f_n = f_{n-1} - 8.5 ) for ( n geq 2 )

a. is this sequence arithmetic, geometric, or neither? why?
b. list at least the first five terms of the sequence.
c. graph the value of the term ( f_n ) as a function of the term number ( n ) for at least the first five terms of the sequence.
graph grid: x - axis (term number) from 1 to 6, y - axis (value) from 0 to 60 with grid lines

Explanation:

Part a:

Step1: Recall sequence types

An arithmetic sequence has a common difference \(d\) (constant difference between consecutive terms). A geometric sequence has a common ratio \(r\) (constant ratio between consecutive terms). The recursive formula is \(f_1 = 60\), \(f_n=f_{n - 1}-8.5\) for \(n\geq2\).

Step2: Check differences/ratios

For consecutive terms, the difference \(f_n - f_{n - 1}=- 8.5\) (constant). For a geometric sequence, the ratio \(\frac{f_n}{f_{n - 1}}\) would need to be constant, but here we subtract a constant, not multiply. So it's arithmetic because there's a common difference.

Step1: Find \(f_1\)

Given \(f_1 = 60\).

Step2: Find \(f_2\)

Using \(f_n=f_{n - 1}-8.5\) with \(n = 2\), \(f_2=f_1 - 8.5=60 - 8.5 = 51.5\).

Step3: Find \(f_3\)

For \(n = 3\), \(f_3=f_2 - 8.5=51.5 - 8.5 = 43\).

Step4: Find \(f_4\)

For \(n = 4\), \(f_4=f_3 - 8.5=43 - 8.5 = 34.5\).

Step5: Find \(f_5\)

For \(n = 5\), \(f_5=f_4 - 8.5=34.5 - 8.5 = 26\).

Answer:

This sequence is arithmetic. Because the recursive formula shows that each term is obtained by subtracting a constant (\(8.5\)) from the previous term, meaning there is a common difference (\(d=-8.5\)).

Part b: